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ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import Mathlib.Analysis.VonNeumannAlgebra.Basic
import Mathlib.Analysis.InnerProductSpace.Adjoint
namespace OAI
noncomputable section
namespace FiniteFactor
open scoped Topology
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
/-- A concrete finite tracial von Neumann algebra in a faithful vector-trace
representation. These are operator-algebra data, without approximation assumptions. -/
structure TracialRepresentation (M : VonNeumannAlgebra H) where
vector : H
norm_vector : ‖vector‖ = 1
cyclic : ∀ S T : M.toStarSubalgebra,
inner ℂ vector ((S : H →L[ℂ] H) ((T : H →L[ℂ] H) vector)) =
inner ℂ vector ((T : H →L[ℂ] H) ((S : H →L[ℂ] H) vector))
separating : ∀ T : M.toStarSubalgebra, (T : H →L[ℂ] H) vector = 0 → T = 0
namespace TracialRepresentation
variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M)
abbrev Algebra := M.toStarSubalgebra
def trace (T : Algebra (M := M)) : ℂ := inner ℂ τ.vector ((T : H →L[ℂ] H) τ.vector)
def norm₂ (T : Algebra (M := M)) : ℝ := ‖(T : H →L[ℂ] H) τ.vector‖
@[simp] theorem trace_one : τ.trace 1 = 1 := by
simp [trace, inner_self_eq_norm_sq_to_K, τ.norm_vector]
theorem trace_mul_comm (S T : Algebra (M := M)) : τ.trace (S*T) = τ.trace (T*S) :=
τ.cyclic S T
theorem trace_star_mul (T : Algebra (M := M)) :
τ.trace (star T*T) = (τ.norm₂ T : ℂ)^2 := by
unfold trace norm₂
change inner ℂ τ.vector ((star (T : H →L[ℂ] H)) ((T : H →L[ℂ] H) τ.vector)) = _
rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_right, inner_self_eq_norm_sq_to_K]
norm_cast
theorem norm₂_star (T : Algebra (M := M)) : τ.norm₂ (star T) = τ.norm₂ T := by
have h := τ.trace_mul_comm (star T) T
have h2 := τ.trace_star_mul (star T)
simp only [star_star] at h2
rw [τ.trace_star_mul, h2] at h
have hr := congrArg Complex.re h
simp only [← Complex.ofReal_pow, Complex.ofReal_re] at hr
have hn : 0 ≤ τ.norm₂ T := norm_nonneg _
have hs : 0 ≤ τ.norm₂ (star T) := norm_nonneg _
nlinarith
@[simp] theorem norm₂_zero : τ.norm₂ 0 = 0 := by simp [norm₂]
@[simp] theorem norm₂_sub (S T : Algebra (M := M)) :
τ.norm₂ (S-T) = ‖(S : H →L[ℂ] H) τ.vector - (T : H →L[ℂ] H) τ.vector‖ := rfl
theorem norm₂_add_le (S T : Algebra (M := M)) : τ.norm₂ (S+T) ≤ τ.norm₂ S+τ.norm₂ T :=
norm_add_le _ _
theorem norm₂_smul (c : ℂ) (T : Algebra (M := M)) : τ.norm₂ (c • T) = ‖c‖*τ.norm₂ T :=
norm_smul _ _
theorem norm₂_mul_le (S T : Algebra (M := M)) : τ.norm₂ (S*T) ≤ ‖S‖ * τ.norm₂ T :=
(S : H →L[ℂ] H).le_opNorm ((T : H →L[ℂ] H) τ.vector)
theorem norm₂_mul_le_right (S T : Algebra (M := M)) : τ.norm₂ (S*T) ≤ τ.norm₂ S * ‖T‖ := by
rw [← τ.norm₂_star (S*T), star_mul]
have h := τ.norm₂_mul_le (star T) (star S)
simpa [τ.norm₂_star, mul_comm, ContinuousLinearMap.star_eq_adjoint] using h
theorem norm₂_le_norm (T : Algebra (M := M)) : τ.norm₂ T ≤ ‖T‖ := by
have h := (T : H →L[ℂ] H).le_opNorm τ.vector
simpa [norm₂, τ.norm_vector] using h
theorem norm_trace_le_norm₂ (T : Algebra (M := M)) : ‖τ.trace T‖ ≤ τ.norm₂ T := by
simpa [trace, norm₂, τ.norm_vector] using norm_inner_le_norm τ.vector ((T : H →L[ℂ] H) τ.vector)
theorem norm₂_eq_zero_iff (T : Algebra (M := M)) : τ.norm₂ T = 0 ↔ T = 0 := by
constructor
· intro h
exact τ.separating T (norm_eq_zero.mp h)
· rintro rfl
exact τ.norm₂_zero
end TracialRepresentation
end FiniteFactor
end
end OAI
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